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An oceanographic tracer. Nuclear weapons tests in the 1950 s and 1960 s released significant tritium (\(_{ 1 }^{ 3 }{H},\) half-life 12.3 years) into the atmosphere. The tritium atoms were quickly bound into water molecules and rained out of the air, most of them ending up in the ocean. For any of this tritium-tagged water that sinks below the surface, the amount of time during which it has been isolated from the surface can be calculated by measuring the ratio of the decay product, \(_{2}^{3} \mathrm{He},\) to the remaining tritium in the water. For example, if the ratio of \(\frac{3}{2} \mathrm{He}\) to \(_{1}^{3} \mathrm{H}\) in a sample of water is \(1 : 1,\) the water has been below the surface for one half-life, or approximately 12 years. This method has provided oceanographers with a convenient way to trace the movements of subsurface currents in parts of the ocean. Suppose that in a particular sample of water, the ratio of \(^{3} \mathrm{He}\) to \(^{3} \mathrm{H}\) is 4.3 to \(1.0 .\) How many years ago did this water sink below the surface?

Step by step solution

01

Understanding the concept

The exercise involves tritium (\(^{3}\mathrm{H}\)) undergoing decay to form helium-3 (\(^{3}\mathrm{He}\)). A ratio between these elements allows us to determine how many half-lives have expired since the water sank below the surface. Each half-life represents 12.3 years.

02

Analyzing the helium to tritium ratio

In this problem, we are given that the helium-3 to tritium ratio is 4.3:1. This means for every 1 unit of remaining tritium, we have 4.3 units of helium-3.

03

Relating the ratio to half-lives

A ratio of 1:1 indicates one half-life. In our case, with a ratio of 4.3:1, calculate how many half-lives have expired by setting up a comparison of the decay process.

04

Equation set-up and solving

Each half-life, the amount of tritium reduces to half, and a corresponding amount of helium-3 is produced. For a 4.3:1 ratio, calculate the equivalent number of half-lives. The amount of helium-3 produced corresponds to \(n\) where \(n\) is the number of half-lives: \[ 2^n - 1 = 4.3 \]

05

Calculating number of half-lives

We solve the equation \(2^n - 1 = 4.3\). Add 1 to both sides to get \(2^n = 5.3\). Take the logarithm: \[ n = \frac{\log(5.3)}{\log(2)} \approx 2.405 \] This results in approximately 2.405 half-lives.

06

Converting half-lives to years

The calculated 2.405 half-lives need to be converted to years. Since one half-life is 12.3 years:\[ 2.405 \times 12.3 \approx 29.5915 \] Therefore, the water sank approximately 29.6 years ago.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Oceanographic tracer

The concept of oceanographic tracers is fascinating as it allows scientists to track and study the intricate movements and behaviors of ocean currents. Such tracers are often chemical substances that help in identifying the origin or the pathway of water masses. Tritium, released during nuclear weapons tests of the 1950s and 60s, acted as an effective tracer. When tritium becomes part of ocean water, its decay product, helium-3, can be measured to gain insights into how and where the water has moved over time. By examining the presence and concentration ratio of tritium and helium-3, researchers can infer the timeline of these movements, giving valuable information about the underwater currents.

Half-life calculation

A half-life is a measure of the time required for a quantity to reduce to half its initial amount. For tritium, approximated at 12.3 years, this concept is crucial for calculating how long the tritium has been decaying in an isolated environment. The relationship between tritium and helium-3, its decay product, gives a direct measure of elapsed time since isolation. By using the ratio of existent helium-3 to remaining tritium, we can determine how many half-life cycles have passed, thus estimating the water's submersion time below the ocean surface. This calculation exploits the exponential nature of radioactive decay and requires understanding of logarithmic functions to solve for elapsed time in real-world applications.

Helium-3 analysis

Helium-3 ( Helium-3) is the decay product of tritium and acts as an excellent marker for calculating radioactive decay over time. Analyzing helium-3 is integral to estimating the duration that tritium-contaminated water has been isolated from the surface. When tritium decays, it converts into this isotope, and the quantity of helium-3 grows proportional to the decay of tritium. By evaluating the ratio of helium-3 to tritium, oceanographers can calculate the elapsed time since the water mass was last exposed to the surface, revealing climatic patterns or changes in ocean currents.

Nuclear weapons tests

The nuclear weapon tests conducted in the mid-20th century had far-reaching effects beyond just political and military implications. During these tests, significant amounts of tritium entered the atmosphere, eventually infiltrating bodies of water like the oceans. This release inadvertently created a means for tracing the movement of ocean waters. As tritium is incorporated into water molecules and rains down, it serves as a wide-reaching natural experiment. Over decades, researchers have been able to leverage this global introduction of tritium to study oceanographic patterns, providing vital data on ocean dynamics that were otherwise difficult to observe.